10 Parametric Equations and Polar Coordinates Copyright Cengage

10 Parametric Equations and Polar Coordinates Copyright © Cengage Learning. All rights reserved.

10. 4 Areas and Lengths in Polar Coordinates Copyright © Cengage Learning. All rights reserved.

Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the area of a sector of a circle: A = r 2 where, as in Figure 1, r is the radius and is the radian measure of the central angle. Figure 1 3

Areas and Lengths in Polar Coordinates Formula 1 follows from the fact that the area of a sector is proportional to its central angle: A = ( /2 ) r 2 = r 2. 4

Areas and Lengths in Polar Coordinates Let be the region, illustrated in Figure 2, bounded by the polar curve r = f ( ) and by the rays = a and = b, where f is a positive continuous function and where 0 < b – a 2. Figure 2 We divide the interval [a, b] into subintervals with endpoints 0, 1, 2, . . . , n and equal width . 5

Areas and Lengths in Polar Coordinates The rays = i then divide into n smaller regions with central angle = i – 1. If we choose in the ith subinterval [ i – 1, i], then the area Ai of the ith region is approximated by the area of the sector of a circle with central angle and radius f ( ). (See Figure 3. ) Figure 3 6

Areas and Lengths in Polar Coordinates Thus from Formula 1 we have Ai [f ( )]2 and so an approximation to the total area A of is It appears from Figure 3 that the approximation in (2) improves as n . 7

Areas and Lengths in Polar Coordinates But the sums in (2) are Riemann sums for the function g( ) = [f ( )]2, so It therefore appears plausible that the formula for the area A of the polar region is 8

Areas and Lengths in Polar Coordinates Formula 3 is often written as with the understanding that r = f ( ). Note the similarity between Formulas 1 and 4. When we apply Formula 3 or 4 it is helpful to think of the area as being swept out by a rotating ray through O that starts with angle a and ends with angle b. 9

Example 1 Find the area enclosed by one loop of the four-leaved rose r = cos 2. Solution: Notice from Figure 4 that the region enclosed by the right loop is swept out by a ray that rotates from = – /4 to = /4. Figure 4 10

Example 1 – Solution cont’d Therefore Formula 4 gives 11

Example 1 – Solution cont’d 12

Arc Length 13

Arc Length To find the length of a polar curve r = f ( ), a b, we regard as a parameter and write the parametric equations of the curve as x = r cos = f ( ) cos y = r sin = f ( ) sin Using the Product Rule and differentiating with respect to , we obtain 14

Arc Length So, using cos 2 + sin 2 = 1, we have 15

Arc Length Assuming that f is continuous, we can write the arc length as Therefore the length of a curve with polar equation r = f ( ), a b, is 16

Example 4 Find the length of the cardioid r = 1 + sin . Solution: The cardioid is shown in Figure 8. r = 1 + sin Figure 8 17

Example 4 – Solution cont’d Its full length is given by the parameter interval 0 2 , so Formula 5 gives 18

Example 4 – Solution cont’d We could evaluate this integral by multiplying and dividing the integrand by , or we could use a computer algebra system. In any event, we find that the length of the cardioid is L = 8. 19